The effect of possible length contraction perpendicular to the direction of motion. Assume that objects shrink perpendicular to the direction of motion. A and B are the same height when both are at rest. Both hold swords parallel to the ground, and B moves past. If B shrinks,A's sword will miss B, but B's sword will cut A. However, from B's point of view, as shown in the right figure,A is moving, and it is A who shrinks.That would result in an injury to B not A.
Each has a sword held out at the level of the top of the head. Now person B is carried past person A at a high speed. According to A, B is moving, and B gets shorter. B's sword cuts A's head while A's sword passes safely over B. According to B, A is moving and the situation is reversed. We have a true contradiction. Each person is wounded in their own rest frame but not in the other. The only way out of this is to say that there can be no length change perpendicular to the direction of motion, and no one gets hurt. (A similar argument would rule out expansion perpendicular to the direction of motion as well as contraction.)
We can think of no such examples to rule out changes parallel to the direction of motion. Here, there is actually a change of length. Moving objects appear to shrink. We call this effect Lorentz contraction. To see this we use Fig. 7.7 to show how we might measure the length of a moving object. The length of an object, measured in the frame in which it is at rest, is called the proper length, L0 . This can be measured in the usual way, since its ends are not going anywhere. We now measure its length in a frame in which it is moving. We can tell its speed v by having two markers at rest in our frame, and measuring the time for the front of the object to travel from one marker to the other. We can then measure its length by
The effect of length changes parallel to the direction of motion. (a) To measure the length of a stick, we must first measure its speed.We do this by measuring the time for one point on the stick to go a known distance between two stationary clocks. In the upper frame, the front of the stick starts at the right clock. In the lower frame it reaches the left clock.The time difference is noted, and the speed is calculated. (b) Knowing the speed of the stick, we measure its length by seeing how long it takes the stick to pass a single stationary clock. In the top half of the frame the front of the stick is at the clock, and the measurement starts. In the bottom half of the frame, the back of the stick reaches the clock and the measurement ends.
timing the passage of the object past one marker. The time interval between the two measurements at the one marker At, as measured in the frame of the object, is
In our frame the time interval is different because of time dilation, so the interval is
We now say that the length of the object is
Finally, substituting At = L0/v gives
Not surprisingly, the length contraction has the same dependence on v/c as does the time dilation.
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